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A 45.0 -kg woman stands up in a 60.0 $\mathrm{kg}$ canoe 5.00 $\mathrm{m}$ long. She walks from a point 1.00 $\mathrm{m}$ from one end to a point 1.00 $\mathrm{m}$ from the other end (Fig. 8.48$) .$ If you ignore resistance to motion of the canoe in the water, how far does the canoe move during thisprocess?

$1.29 \mathrm{m}$ to the left

Physics 101 Mechanics

Chapter 8

Momentum, Impulse, and Collisions

Moment, Impulse, and Collisions

Rutgers, The State University of New Jersey

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once again welcome to a new problem. This time we have a woman, and she's standing on a canoe And the, um, people, the actual length of the canoe happens to be 5 m. So she is standing, um, you know, 11 m from from the left. So this is this is 1 m from the left. Uh huh. And then she's gonna walk towards the right up until our position 1 m from the second edge of the canoe. So in between, we have 3 m, and so she's going to be walking in that direction in the right, Uh, so we'll assume the positive is the right. And so as that happens, as she, um, walks towards the right, what's going to happen is the canoe will move towards the left. So, you know, eventually this is this is kind of what you're gonna be seeing. It's gonna move towards the left, and it's it's gonna move a specific distance. Uh, that's the distance we want to find out. We want to find out what this distance is right here. We're gonna call it. Um um I'm gonna call it a D. Thank you. Uh huh. D canoe and then remember, the woman is also moving forward. So this distance right here is the distance traveled by the woman. Uh, and we want to find out what? So this is the information. The information that's given is that the mass of the woman happens to be 45 kg and also the mass of the man who happens to be 60 kg. That's the information given, uh, the total length, as you can see of the canoe. Uh, this whole land right here mhm is 5 m. Okay, so the length of the canoe is 5 m. That's the information we're given. So we're gonna use the law of conservation of momentum in this problem. And we want to find, uh, the distance that the canoe moves with the with a woman inside of it. So that's what we want to find. Uh, and this is the information that's given. So in terms of a law of conservation, of momentum, the initial momentum, it calls to the final momentum. Initially, both the the woman and the canoe or not moving so would say that the initial velocity of the woman is 0 m per second and then The initial velocity of the canoe is also, um, 0 m per second. We don't know what the final velocity of the, uh, canoe and woman is. Okay, We don't know what that final lost is going to be, Uh, but it's going to be dependent on this distance right here. So, you know, remember the last e is distance over time. So there's a timeframe happening where the canoe and the and the woman are moving. Okay, Uh, so C n w you know, we don't know what that velocity is. So coming back to the law of conservation of momentum, uh, we get to see, we get to see that the initial lost the initial momentum. Uh, the woman is the mass of the woman times high initial velocity. And then the initial momentum of the canoe, uh, is the mass of the canoe. Um, and the initial momentum, um, and the velocity initial velocity of the economy. This is equal to the final loss to remember. The the canoe and the woman are moving towards the left. So it means that their total mass times the combined velocity, the velocity will be negative. That's why we have a minus there. And then you, um you put, uh, that is going to be equal to remember, also at the final part. The woman is moving forward, so she has her own final velocity. Okay, this is final velocity, I think I think we have to change these variables a little bit. Okay, so, um, this one, Yes, this one is the final velocity of the canoe and the woman, and it's negative. Um, and then we also want to find the final velocity of the woman herself because she is moving to the right. So it's positive. Uh, the initial momentum of the system is zero because they're not moving. So this is gonna be zero equals two negative mass of, uh, canoe mas or woman. Uh, final velocity of both of them, plus final momentum of the woman our goal is to solve for at this distance, the sea. Yeah. And so we can shift things around and have the mass of the woman. And the final velocity of the woman equals two, uh, m c plus M w final lost of woman and canoe like that. So they are combined their combined together. Well, actually, you know what? We can actually simplify this problem by saying the the Mm hmm. The Instead of having the woman combined here, we could just have the the country itself. So we could just have the final, uh, final velocity of the we don't have to include, uh, the woman. I think there's an understanding thereof of what's going on. Um, right here because it's combined. So the canoe is carrying both of them. Okay, So even this one right here, this is the final. So we just want to make sure that our variables are right. So this is the final velocity of the canoe. Uh, this is the final velocity of the canoe, and that's the final blast if they can. So that's what we have at the moment. On the next page, we're gonna complete the problem. So what we're saying is that the mass of the woman and the final velocity of the woman equals to the combined mass of the canoe and the woman with the final velocity of the canoe. Remember, The last E for the woman is a distance. She traveled over her time. And then the velocity of the canoe is the distance. The canoe traveled over time. That's those are the basic definitions that we have right there. And so mass of woman distance traveled by a woman of the time. Remember, the woman is traveling this way, and the canoe is traveling that way. So this is the distance traveled by the woman. And then this is the distance traveled by the canoe and we're trying to get the distance Struggled by the end of this one right here. And so, uh, this one equals to mass of canoe plus massive woman, uh, distance of canoe. Over time, we're gonna multiply both sides by the time to get rid of it. And so this cancels out, that cancels out. And so we have a formula for the distance traveled by the canoe, and that happens to be the mass of the woman distance traveled by the woman all over the mass of the canoe, plus the distance, the mass of the woman. So the mass of the woman happens to be 45 kilograms. Mm. And then the distance struggled by the woman were given that as if you go back, you see, it happens to be, uh, 3 m we're gonna have 3 m right there and then divided by the mass of the canoe is 60 kilograms. Uh, and then the my the plus, we want to add it. We want to add the numbers. So there's a plus in between, plus the mass of the woman, which is 45 kg. Mhm. And so when we simplify that problem, we'll get the distance. Struggled by the canoe becomes 1.285 m. Hope you enjoyed the problem. Feel free to ask any questions or send comments my way. Uh, and have a wonderful day. Okay, thanks. Bye.

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